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The modular group $\mathrm{PSL}_2(\mathbb{Z})$ has 3 generators $A,B,C$, where $$A:z\to z+1,\quad B:z\to z-1,\quad C:z\to -1/z.$$

I want to compute the automaton that recognize the words of the modular group, where the input symbols are just $\{A,B,C\}$. But the above representation is not in the standard form of a coxeter group system.

My problem is: how can one do the computation in this case? Or are there any programs that do the work available?

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The Warwick automatic groups software provides the means for doing this (in particular, the KBMAG program). Alternatively you can use Alun William's MAF (Monoid Automata Factory) package.

There is also an interface to KBMAG provided in the GAP computer algebra system.

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